Essential Calculus Concepts: Limits, Derivatives, and Integrals
School
University of California, Davis**We aren't endorsed by this school
Course
MAT 22
Subject
Mathematics
Date
Dec 11, 2024
Pages
2
Uploaded by LieutenantReindeerMaster92
1. Limit Calculation:Evaluate the limit:limx→0sin(x)x\lim_{x \to 0} \frac{\sin(x)}{x}x→0limxsin(x) 2. Derivative of a Function:Find the derivative of the following function:f(x)=3x4−5x3+2x−1f(x) = 3x^4 - 5x^3 + 2x - 1f(x)=3x4−5x3+2x−13. Chain Rule:Differentiate the following function using the chain rule:g(x)=sin(5x2+3)g(x) = \sin(5x^2 + 3)g(x)=sin(5x2+3)4. Product Rule:Differentiate the product of two functions:h(x)=(2x3+x)(5x2−4)h(x) = (2x^3 + x)(5x^2 - 4)h(x)=(2x3+x)(5x2−4)5. Integral Calculation (Definite Integral):Compute the following definite integral:∫02(4x2−3x+1) dx\int_0^2 (4x^2 - 3x + 1) \, dx∫02(4x2−3x+1)dx6. Integral Using Substitution:Evaluate the following integral using substitution:∫(2x⋅ex2) dx\int (2x \cdot e^{x^2}) \, dx∫(2x⋅ex2)dx7. Second Derivative:Find the second derivative of the function:f(x)=4x3−3x2+2x−7f(x) = 4x^3 - 3x^2 + 2x - 7f(x)=4x3−3x2+2x−78. Implicit Differentiation:Differentiate the following equation implicitly:
x2+y2=25x^2 + y^2 = 25x2+y2=259. Application of Derivatives (Maximum/Minimum):Find the critical points and determine if they are local maxima or minima for the function:f(x)=x3−6x2+9xf(x) = x^3 - 6x^2 + 9xf(x)=x3−6x2+9x10. Area Between Curves:Find the area between the curves y=x2y = x^2y=x2 and y=4y = 4y=4 from x=−2x = -2x=−2 to x=2x = 2x=2.